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A Fractional-Order Predator–Prey Model with Ratio-Dependent Functional Response and Linear Harvesting

Author

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  • Agus Suryanto

    (Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang 65145, Indonesia)

  • Isnani Darti

    (Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang 65145, Indonesia)

  • Hasan S. Panigoro

    (Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang 65145, Indonesia
    Department of Mathematics, Faculty of Mathematics and Natural Sciences, State University of Gorontalo, Gorontalo 96128, Indonesia)

  • Adem Kilicman

    (Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia)

Abstract

We consider a model of predator–prey interaction at fractional-order where the predation obeys the ratio-dependent functional response and the prey is linearly harvested. For the proposed model, we show the existence, uniqueness, non-negativity and boundedness of the solutions. Conditions for the existence of all possible equilibrium points and their stability criteria, both locally and globally, are also investigated. The local stability conditions are derived using the Magtinon’s theorem, while the global stability is proven by formulating an appropriate Lyapunov function. The occurrence of Hopf bifurcation around the interior point is also shown analytically. At the end, we implemented the Predictor–Corrector scheme to perform some numerical simulations.

Suggested Citation

  • Agus Suryanto & Isnani Darti & Hasan S. Panigoro & Adem Kilicman, 2019. "A Fractional-Order Predator–Prey Model with Ratio-Dependent Functional Response and Linear Harvesting," Mathematics, MDPI, vol. 7(11), pages 1-13, November.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:11:p:1100-:d:286750
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    References listed on IDEAS

    as
    1. Hamdan, Nur ’Izzati & Kilicman, Adem, 2018. "A fractional order SIR epidemic model for dengue transmission," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 55-62.
    2. Moustafa, Mahmoud & Mohd, Mohd Hafiz & Ismail, Ahmad Izani & Abdullah, Farah Aini, 2018. "Dynamical analysis of a fractional-order Rosenzweig–MacArthur model incorporating a prey refuge," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 1-13.
    3. Sung Kyu Choi & Bowon Kang & Namjip Koo, 2014. "Stability for Caputo Fractional Differential Systems," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-6, January.
    4. Agus Suryanto & Isnani Darti & Syaiful Anam, 2017. "Stability Analysis of a Fractional Order Modified Leslie-Gower Model with Additive Allee Effect," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2017, pages 1-9, May.
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    Cited by:

    1. Uddin, Md. Jasim & Rana, Sarker Md. Sohel & Işık, Seval & Kangalgil, Figen, 2023. "On the qualitative study of a discrete fractional order prey–predator model with the effects of harvesting on predator population," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).

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